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A domain wall of relative phase in a flattened harmonically-trapped Bose-Einstein condensed mixture of two atomic hyperfine states, subject to a stationary Rabi coupling of intensity $\Omega$, is predicted to decay through two different mechanisms. For small values of $\Omega$ the instability has an energetic nature and is associated with the formation of a vortex-antivortex pair of the same atomic hyperfine states, whose motion inside the trap causes the emergence of magnetization, the bending of the domain wall and its consequent fragmentation. For large values of $\Omega$ the domain wall instead undergoes a dynamic snake instability, caused by the negative value of its effective mass and results in the fast fragmentation of the wall into smaller domain walls confining vortex pairs of different atomic species. Numerical predictions are given by solving the time-dependent Gross-Pitaevskii equation in experimentally available configurations of mixtures of sodium atomic gases., Comment: 9 pages (5 of main text and 4 of supplementary material), 4 figures

We provide an overview of the recent theoretical and experimental advances in the study of second sound in ultracold atomic gases. Starting from the Landau two fluid hydrodynamic equations we develop the theory of first and second sound in various configurations characterized by different geometries and quantum statistics. These include the weakly interacting 3D Bose gas, the strongly interacting Fermi gas at unitarity in the presence of highly elongated traps and the dilute 2D Bose gas, characterized by the Berezinskii-Kosterlitz-Thouless transition. An explicit comparison with the propagation of second sound in liquid Helium is carried out to elucidate the main analogies and differences. We also make an explicit comparison with the available experimental data and point out the crucial role played by the superfluid density in determining the temperature dependence of the second sound speed.

We analyse, theoretically and experimentally, the nature of solitonic vortices (SV) in an elongated Bose-Einstein condensate. In the experiment, such defects are created via the Kibble-Zurek mechanism, when the temperature of a gas of sodium atoms is quenched across the BEC transition, and are imaged after a free expansion of the condensate. By using the Gross-Pitaevskii equation, we calculate the in-trap density and phase distributions characterizing a SV in the crossover from an elongate quasi-1D to a bulk 3D regime. The simulations show that the free expansion strongly amplifies the key features of a SV and produces a remarkable twist of the solitonic plane due to the quantized vorticity associated with the defect. Good agreement is found between simulations and experiments., Comment: 6 pages, 4 figures

Using linear response theory within the Random Phase Approximation, we investigate the propagation of sound in a uniform two dimensional (2D) Bose gas in the collisionless regime. We show that the sudden removal of a static density perturbation produces a damped oscillatory behavior revealing that sound can propagate also in the absence of collisions, due to mean-field interaction effects. Our analysis points out the crucial role played by Landau damping. We support our predictions by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii equation. The results are consistent with the recent experimental observation of sound in a weakly interacting 2D Bose gas both below and above the superfluid Berezinskii-Kosterlitz-Thouless transition., Main text: 4 pages, 4 figures + Supplementary material: 2 page. The general structure has been revised for clarity, with improved introduction and conclusions. Accepted for publication in Physical Review Letters

When the spectrum of magnetic excitations of a quantum mixture is much softer than the density spectrum, the system becomes effectively incompressible and can host magnetic defects. These are characterized by the presence of a topological defect in one of the two species and by a local modification of the density in the second one, the total density being practically unaffected. For miscible mixtures interacting with equal intraspecies coupling constants the width of these magnetic defects is fixed by the difference between the intraspecies and interspecies coupling constants and becomes larger and larger as one approaches the demixing transition. When the density of the filling component decreases, the incompressibility condition breaks down and we predict the existence of a critical filling, below which all the atoms of the minority component remain bound in the core of the topological defect. Applications to the sodium case both in uniform and harmonically trapped configurations are considered and a protocol to produce experimentally these defects is discussed. The case of binary mixtures interacting with unequal intraspecies forces and experiencing buoyancy is also addressed., Comment: 10 pages, 6 figures

Superfluidity is a macroscopic quantum phenomenon occurring in systems as diverse as liquid helium and neutron stars. It occurs below a critical temperature(1,2) and leads to peculiar behaviour such as frictionless flow, the formation of quantized vortices and quenching of the moment of inertia. Ultracold atomic gases offer control of interactions and external confinement, providing unique opportunities to explore superfluid phenomena. Many such (finite-temperature) phenomena can be explained in terms of a two-fluid mixture(3,4) comprising a normal component, which behaves like an ordinary fluid, and a superfluid component with zero viscosity and zero entropy. The two-component nature of a superfluid is manifest in 'second sound', an entropy wave in which the superfluid and the non-superfluid components oscillate with opposite phases (as opposed to ordinary 'first sound', where they oscillate in phase). Here we report the observation of second sound in an ultracold Fermi gas with resonant interactions. The speed of second sound depends explicitly on the value of the superfluid fraction(5), a quantity that is sensitive to the spectrum of elementary excitations(6). Our measurements allow us to extract the temperature dependence of the superfluid fraction, a previously inaccessible quantity that will provide a benchmark for theories of strongly interacting quantum gases.

By developing a hydrodynamic formalism, we investigate the expansion dynamics of the single-minimum phase of a binary spin-orbit coupled Bose-Einstein condensate, after releasing from an external harmonic trap. We find that the expansion of the condensate along the direction of the spin-orbit coupling is dramatically slowed down near the transition between the single-minimum phase and the plane-wave phase. Such a slow expansion, resembling a form of an effective localization, is due to the quenching of the superfluid motion which results in a strong increase of the effective mass. In the single-minimum phase the anisotropic expansion of the Bose gas, which is spin balanced at equilibrium, is accompanied by the emergence of a local spin polarization. Our analytic scaling solutions emerging from hydrodynamic picture are compared with a full numerical simulation based on the coupled Gross-Pitaevskii equations., 5 pages, 3 figures

A variety of solitary waves, such as solitons, vortex rings, solitonic vortices, and more complex entities, have recently been predicted to exist. They can move in superfluid ultracold gases along elongated traps. The theoretical description of this motion requires knowledge of the inertial soliton mass and the effective number of particles in it as functions of the soliton energy. While these functions can be calculated by a microscopic theory, it is also possible to express them directly in terms of observable quantities, such as the order parameter phase jump and the particle number depletion in the soliton. In this article, the corresponding equations are derived in a simple and physically clear way and applied to the recently predicted `magnetic soliton' in mixtures of Bose gases in various spin states., Comment: 7 pages. Published in the special October 2016 issue of Uspekhi Fizicheskikh Nauk [Physics-Uspekhi] journal on the occasion of the 100th anniversary of the birth of V. L. Ginzburg

We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the $1/k^4$ tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of Chang \textit{et al.} [Phys. Rev. Lett. \textbf{117}, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable $^4$He atoms, where a $1/r^4$ tail in the density distribution was observed after a large expansion time, implying the existence of the $1/k^4$ tail in the asymptotic momentum distribution., published version, 9 pages, 5 figures

This chapter discusses the magnetic properties of interacting Fermi gases. It considers the transition to the ferromagnetic phase (itinerant ferromagnetism) of a repulsive Fermi gas as a consequence of the Stoner instability. The dependence of the magnetic susceptibility on the relevant interaction coupling constant is explicitly discussed. The chapter then investigates the competition between superfluidity and ferromagnetism and the nature of the Clogston–Chandrasekhar transition, both in uniform and harmonically trapped Fermi gases, with special focus on the behaviour at unitarity, where systematic experimental data are available. It finally addresses the problem of the polaron in highly polarized configurations and the transition from the polaron to molecular binding along the BCS–BEC crossover.

We analyze theoretically the quantization of conductance occurring with cold bosonic atoms trapped in two reservoirs connected by a constriction with an attractive gate potential. We focus on temperatures slightly above the condensation threshold in the reservoirs. We show that a conductance step occurs, coinciding with the appearance of a condensate in the constriction. Conductance relies on a collective process involving the quantum condensation of an atom into an elementary excitation and the subsequent quantum evaporation of an atom, in contrast with ballistic fermion transport. The value of the bosonic conductance plateau is strongly enhanced compared to fermions and explicitly depends on temperature. We highlight the role of the repulsive interactions between the bosons in preventing them from collapsing into the constriction. We also point out the differences between the bosonic and fermionic thermoelectric effects in the quantized conductance regime.

We review generally accepted definitions of Bose–Einstein condensation and superfluidity, emphasizing that they are independent concepts. These ideas are illustrated in a dilute hard-sphere Bose gas, which is relevant to experiments on excitons and spin-aligned atomic hydrogen. We then discuss superfluid He in porous media, as simulated by different models in different regimes. At low coverage, we model it by a dilute hard-sphere Bose gas in random potentials, and show that superfluidity is destroyed through the pinning of the Bose condensate by the external potentials. At full coverage, we model the random medium by an ohmic network of random resistors, and argue that the superfluid transition is a percolation transition in d = 3, with critical exponent 1.7. This book is devoted to the phenomenon of Bose–Einstein condensation [1, 2] and inevitably, its relevance to superfluidity [3]. To provide some background for other articles in this volume, I would like to summarize some commonly accepted views on these phenomena, and illustrate them in the context of a dilute hard-sphere Bose gas, a model in which we have some control over the approximations made. I will also describe some recent work on the effect of randomness on the Bose condensate, which shows that Bose–Einstein condensation does not automatically give rise to superfluidity.

This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.

This chapter develops the formalism of linear response function, dynamic structure factor, dynamic polarizability, and sum rules. The behaviour of the density response function in a superfluid is discussed and analytic results at finite temperatures are shown in the case of the ideal and weakly interacting Bose gases. Sum rules are used to derive general inequalities holding at both zero and finite temperatures. The Hohenberg–Mermin–Wagner theorem and its generalization to zero temperature, based on the use of the uncertainty inequality, are explicitly discussed in the case of Bose systems.

We study solitary waves of polarization (magnetic solitons) in a two-component Bose gas with slightly unequal repulsive intra- and interspin interactions. In experimentally relevant conditions we obtain an analytical solution which reveals that the width and the velocity of magnetic solitons are explicitly related to the spin healing length and the spin sound velocity of the Bose mixture, respectively. We calculate the profiles, the energy and the effective mass of the solitons in the absence of external fields and investigate their oscillation in a harmonic trap where the oscillation period is calculated as a function of the oscillation amplitude. The stability of magnetic solitons in two dimensions and the conditions for their experimental observation are also briefly discussed., published version, 4.5 pages, 4 figures

This chapter develops the mean-field Bogoliubov theory of the weakly interacting Bose–Einstein condensed gas. Results for the ground state energy, momentum distribution, and excitation spectrum are discussed. Emphasis is given to the behaviour of the sound velocity and its relationship with the s-wave scattering length. Special focus is given to the infrared divergent behaviour of the momentum distribution as well as to its behaviour at high momenta. The concept of healing length is introduced. The quantum and thermal depletion of the condensate fraction are explicitly derived. Beyond-mean effects in the equation of state, accounted for by the Lee–Yang–Huang expansion, are also discussed.

The dynamic behavior of vortex pairs in two-component coherently (Rabi) coupled Bose-Einstein condensates is investigated in the presence of harmonic trapping. We discuss the role of the surface tension associated with the domain wall connecting two vortices in condensates of atoms occupying different spin states and its effect on the precession of the vortex pair. The results, based on the numerical solution of the Gross-Pitaevskii equations, are compared with the predictions of an analytical macroscopic model and are discussed as a function of the size of the pair, the Rabi coupling and the inter-component interaction. We show that the increase of the Rabi coupling results in the disintegration of the domain wall into smaller pieces, connecting vortices of new-created vortex pairs. The resulting scenario is the analogue of quark confinement and string breaking in quantum chromodynamics., Comment: 5 pages, 5 figures

We study magnetic solitons, solitary waves of spin polarization (i.e., magnetization), in binary Bose-Einstein condensates in the presence of Rabi coupling. We show that the system exhibits two types of magnetic solitons, called $2\pi$ and $0\pi$ solitons, characterized by a different behavior of the relative phase between the two spin components. $2\pi$ solitons exhibit a $2\pi$ jump of the relative phase, independent of their velocity, the static domain wall explored by Son and Stephanov being an example of such $2\pi$ solitons with vanishing velocity and magnetization. $0\pi$ solitons instead do not exhibit any asymptotic jump in the relative phase. Systematic results are provided for both types of solitons in uniform matter. Numerical calculations in the presence of a one-dimensional harmonic trap reveal that a $2\pi$ soliton evolves in time into a $0\pi$ soliton, and vice versa, oscillating around the center of the trap. Results for the effective mass, the Landau critical velocity, and the role of the transverse confinement are also discussed., Comment: published version, 15 pages, 16 figures

We discuss the superfluid properties of a Bose-Einstein condensed gas with spin-orbit coupling, recently realized in experiments. We find a finite normal fluid density $\rho_n$ at zero temperature which turns out to be a function of the Raman coupling. In particular, the entire fluid becomes normal at the transition point from the zero momentum to the plane wave phase, even though the condensate fraction remains finite. We emphasize the crucial role played by the gapped branch of the elementary excitations and discuss its contributions to various sum rules. Finally, we prove that an independent definition of superfluid density $\rho_s$, using the phase twist method, satisfies the equality $\rho_n+\rho_s=\rho$, the total density, despite the breaking of Galilean invariance.

Using Landau's theory of two-fluid hydrodynamics we investigate the sound modes propagating in a uniform weakly-interacting superfluid Bose gas for values of temperature, up to the critical point. In order to evaluate the relevant thermodynamic functions needed to solve the hydrodynamic equations, including the temperature dependence of the superfluid density, we use Bogoliubov theory at low temperatures and the results of a perturbative approach based on Beliaev diagrammatic technique at higher temperatures. Special focus is given on the hybridization phenomenon between first and second sound which occurs at low temperatures of the order of the interaction energy and we discuss explicitly the behavior of the two sound velocities near the hybridization point., 5 pages, 4 figures

We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (``Bloch bands''). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.

We analyze theoretically the transport properties of a weakly-interacting ultracold Bose gas enclosed in two reservoirs connected by a constriction. We assume that the transport of the superfluid part is hydrodynamic, and we describe the ballistic transport of the normal part using the Landauer-Buttiker formalism. Modeling the coupled evolution of the phase, atom number, and temperature mismatches between the reservoirs, we predict that Helmholtz (plasma) oscillations, induced by an initial imbalance in atom numbers, can be observed at non-zero temperatures below Tc. We show that, because of its strong compressibility, the ultracold Bose gas is characterized by a fast thermalization compared to the damping time for plasma oscillations, accompanied by a fast transfer of the normal component through the constriction. This fast thermalization also affects the gas above Tc, where we present an explicit comparison to the ideal fermionic case., 5 pages, 5 figures

We observe solitonic vortices in an atomic Bose-Einstein condensate after free expansion. Clear signatures of the nature of such defects are the twisted planar density depletion around the vortex line, observed in absorption images, and the double dislocation in the interference pattern obtained through homodyne techniques. Both methods allow us to determine the sign of the quantized circulation. Experimental observations agree with numerical simulations. These solitonic vortices are the decay product of phase defects of the BEC order parameter spontaneously created after a rapid quench across the BEC transition in a cigar-shaped harmonic trap and are shown to have a very long lifetime., 7 pages, 7 figures

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach., revtex, 69 pages, 38 eps figures, new version with more references, new figures, various changes and corrections, for publ. in Rev. Mod. Phys., available also at http://www-phys.science.unitn.it/bec/BEC.html

We consider the influence of an ω-dependent ionic dielectric constant ϵ(ω) on the properties of a superconductor. Assuming that the pairing interaction is proportional to ϵ2 we have solved the Eliashberg equations for this case, both for imaginary and real frequencies. The interaction potential depends on a coupling constant λ and on a longitudinal phonon frequency ω. The dielectric constant is assumed to be independent of wavevector q, and to depend on frequency through the expression: ϵ(ω) = (ω2 - ω2long)/(ω2 - ω2trans), where ωlong, ωtrans are the frequencies of optical phonons of the dielectric. We find that along the imaginary frequency axis (but not for real frequencies) the weighted phonon propagator can be modeled by an appropriate choice of a cutoff frequency and an effective coupling constant. The influence of ϵ(ω) on Tc, the gap Δ(ω), and the renormalization function Z(ω) are studied and it is found that these quantities increase significantly with the dielectric constant.

We study the scattering of atoms, rotons and phonons at the free surface of superfluid4He. We calculate the evaporation, condensation and reflection probabilities in the framework of linearized time dependent density functional theory accounting for elastic one-to-one processes. We explore the angular dependence of the scattering probabilities showing that most results are independent of the incidence angle and can be expressed in terms of a single energy dependent parameter.

This paper is devoted to the calculation of the momentum of localized excitations, such as solitons and vortex rings, moving in a superfluid. The direct calculation of the momentum by integration of the mass flux density results in a badly-converging integral. I suggest a method for the renormalization of the integral with the explicit separation of a term related to the vortex line. This term can be calculated explicitly and gives the main contribution for the rings whose size is large compared to the healing length. I compare my method with the Jones and Roberts prescription for the renormalization. I investigate the case of a uniform superfluid, and that of a superfluid in a cylindrical trap. I discuss the calculation of the jump in the phase of the order parameter and obtain a simple estimate for this jump for a large ring in the trap., Comment: Paper accepted for publication in the special issue of JETP devoted to 75-th anniversary of A.F. Andreev, 10 pages

We investigate the behavior of large amplitude oscillations of a trapped Bose-condensed gas of alkali atoms at zero temperature, by solving the equations of hydrodynamics for collective modes. Due to the atom-atom interaction, the equations of motion are nonlinear and give rise to significant frequency shift and mode coupling. We provide analytic expressions for the frequency shift, pointing out the crucial role played by the anisotropy of the confining potential. For special values of the anisotropy parameter the mode coupling is particularly strong and the frequency shift becomes large, revealing a peculiar behavior of the Bose-condensed gas. Consequences on the theory of collapse and revival of collective excitations are also discussed., 10 pages, RevTeX, 9 figures, more info at http://www-phys.science.unitn.it/bec/BEC.html